Why Monte Carlo epsilon-soft approach cannot compute $\max Q(s,a)$?
If $\pi$ is a random policy, and after running through this algorithm, and for each state take the $\max Q(s,a)$ for all possible actions, why would that not be equal to $Q_{\pi^*}(s, a)$ (optimal Q function)? 


Assuming that the estimates for $Q_{\pi}(s,a)$ have converged to close to correct values from many samples, then a policy based on $\pi'(s) = \text{argmax}_a Q_{\pi}(s,a)$ is not guaranteed to be an optimal policy unless the policy $\pi$ being measured is already the optimal policy.

This is because the action value $Q_{\pi}(s,a)$ gives the expected future reward from taking action $a$ in state $s$, and from that point on following the policy $\pi$. The function does not, by itself, adapt to the idea that you might change other action choices as well. It is a measure of immediate differences between action choices at any given time step. Therefore if there are any long-term dependencies where your action choice at $t$ would be different if only you could guarantee a certain choice at $t+1$ or later, this cannot be resolved by simply taking the maximum $Q_{\pi}(s,a)$ when $\pi$ was a simple random policy.

However, if you do decide to change the policy such that you always follow actions $\pi'(s) = \text{argmax}_a Q_{\pi}(s,a)$ for all states, then you can say this: For each state, $V_{\pi'}(s) \ge V_{\pi}(s)$. I.e.  $\pi'(s)$ is no worse than, and may be a strict improvement over $\pi(s)$. Better than that, $\pi'(s)$ will be a strict improvment over $\pi(s)$ if $Q_{\pi}(s,a)$ is accurate and $\pi(s)$ is not already the optimal policy $\pi^*(s)$. This is the basis for the Policy Improvement Theorem which shows that if you repeat the process of measuring $Q_{\pi^k}$ and then creating a new policy $\pi^{k+1}(s) = \text{argmax}_a Q_{\pi^k}(s,a)$ that you will eventually find the optimal policy. You only have to repeat your idea many times to eventually find $\pi^*$

The Dynamic Programming technique Policy Iteration does this exactly. All other value-based Reinforcement Learning methods are variations of this idea and rely at least in part on this proof.